math.sqrt on complex, real part

Percentage Accurate: 41.9% → 84.3%
Time: 4.5s
Alternatives: 5
Speedup: 2.0×

Specification

?
\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \end{array} \]
(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))
double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
}

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
end function

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
}

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
end

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
end

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 5 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 41.9% accurate, 1.0× speedup?

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \end{array} \]
(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))
double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
}

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
end function

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
}

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
end

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
end

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
\end{array}

Alternative 1: 84.3% accurate, 0.5× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} \cdot -0.5}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]
(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))) 0.0)
   (* 0.5 (sqrt (* 2.0 (/ (* (pow im 2.0) -0.5) re))))
   (sqrt (* 0.5 (+ re (hypot re im))))))
double code(double re, double im) {
	double tmp;
	if (sqrt((2.0 * (re + sqrt(((re * re) + (im * im)))))) <= 0.0) {
		tmp = 0.5 * sqrt((2.0 * ((pow(im, 2.0) * -0.5) / re)));
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	}
	return tmp;
}

public static double code(double re, double im) {
	double tmp;
	if (Math.sqrt((2.0 * (re + Math.sqrt(((re * re) + (im * im)))))) <= 0.0) {
		tmp = 0.5 * Math.sqrt((2.0 * ((Math.pow(im, 2.0) * -0.5) / re)));
	} else {
		tmp = Math.sqrt((0.5 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

def code(re, im):
	tmp = 0
	if math.sqrt((2.0 * (re + math.sqrt(((re * re) + (im * im)))))) <= 0.0:
		tmp = 0.5 * math.sqrt((2.0 * ((math.pow(im, 2.0) * -0.5) / re)))
	else:
		tmp = math.sqrt((0.5 * (re + math.hypot(re, im))))
	return tmp

function code(re, im)
	tmp = 0.0
	if (sqrt(Float64(2.0 * Float64(re + sqrt(Float64(Float64(re * re) + Float64(im * im)))))) <= 0.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64((im ^ 2.0) * -0.5) / re))));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im))));
	end
	return tmp
end

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (sqrt((2.0 * (re + sqrt(((re * re) + (im * im)))))) <= 0.0)
		tmp = 0.5 * sqrt((2.0 * (((im ^ 2.0) * -0.5) / re)));
	else
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

code[re_, im_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(N[Power[im, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} \cdot -0.5}{re}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 2: 79.2% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \end{array} \]
(FPCore (re im) :precision binary64 (sqrt (* 0.5 (+ re (hypot re im)))))
double code(double re, double im) {
	return sqrt((0.5 * (re + hypot(re, im))));
}

public static double code(double re, double im) {
	return Math.sqrt((0.5 * (re + Math.hypot(re, im))));
}

def code(re, im):
	return math.sqrt((0.5 * (re + math.hypot(re, im))))

function code(re, im)
	return sqrt(Float64(0.5 * Float64(re + hypot(re, im))))
end

function tmp = code(re, im)
	tmp = sqrt((0.5 * (re + hypot(re, im))));
end

code[re_, im_] := N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 3: 42.0% accurate, 2.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.3 \cdot 10^{+205}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{elif}\;re \leq 3.8 \cdot 10^{+37}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]
(FPCore (re im)
 :precision binary64
 (if (<= re -2.3e+205)
   (* 0.5 (sqrt (* 2.0 (- re re))))
   (if (<= re 3.8e+37) (* 0.5 (sqrt (* 2.0 im))) (sqrt re))))
double code(double re, double im) {
	double tmp;
	if (re <= -2.3e+205) {
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	} else if (re <= 3.8e+37) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.3d+205)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - re)))
    else if (re <= 3.8d+37) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.3e+205) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - re)));
	} else if (re <= 3.8e+37) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

def code(re, im):
	tmp = 0
	if re <= -2.3e+205:
		tmp = 0.5 * math.sqrt((2.0 * (re - re)))
	elif re <= 3.8e+37:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = math.sqrt(re)
	return tmp

function code(re, im)
	tmp = 0.0
	if (re <= -2.3e+205)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - re))));
	elseif (re <= 3.8e+37)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.3e+205)
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	elseif (re <= 3.8e+37)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

code[re_, im_] := If[LessEqual[re, -2.3e+205], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.8e+37], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;re \leq -2.3 \cdot 10^{+205}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\

\mathbf{elif}\;re \leq 3.8 \cdot 10^{+37}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{re}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 4: 40.8% accurate, 2.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.75 \cdot 10^{+37}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]
(FPCore (re im)
 :precision binary64
 (if (<= re 3.75e+37) (* 0.5 (sqrt (* 2.0 im))) (sqrt re)))
double code(double re, double im) {
	double tmp;
	if (re <= 3.75e+37) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 3.75d+37) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.75e+37) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

def code(re, im):
	tmp = 0
	if re <= 3.75e+37:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = math.sqrt(re)
	return tmp

function code(re, im)
	tmp = 0.0
	if (re <= 3.75e+37)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.75e+37)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

code[re_, im_] := If[LessEqual[re, 3.75e+37], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;re \leq 3.75 \cdot 10^{+37}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{re}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 5: 26.9% accurate, 2.1× speedup?

\[\begin{array}{l} \\ \sqrt{re} \end{array} \]
(FPCore (re im) :precision binary64 (sqrt re))
double code(double re, double im) {
	return sqrt(re);
}

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt(re)
end function

public static double code(double re, double im) {
	return Math.sqrt(re);
}

def code(re, im):
	return math.sqrt(re)

function code(re, im)
	return sqrt(re)
end

function tmp = code(re, im)
	tmp = sqrt(re);
end

code[re_, im_] := N[Sqrt[re], $MachinePrecision]

\begin{array}{l}

\\
\sqrt{re}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Developer target: 48.8% accurate, 0.7× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t_0 - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t_0 + re\right)}\\ \end{array} \end{array} \]
(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* re re) (* im im)))))
   (if (< re 0.0)
     (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- t_0 re)))))
     (* 0.5 (sqrt (* 2.0 (+ t_0 re)))))))
double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((re * re) + (im * im)))
    if (re < 0.0d0) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(((im * im) / (t_0 - re))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (t_0 + re)))
    end if
    code = tmp
end function

public static double code(double re, double im) {
	double t_0 = Math.sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

def code(re, im):
	t_0 = math.sqrt(((re * re) + (im * im)))
	tmp = 0
	if re < 0.0:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(((im * im) / (t_0 - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (t_0 + re)))
	return tmp

function code(re, im)
	t_0 = sqrt(Float64(Float64(re * re) + Float64(im * im)))
	tmp = 0.0
	if (re < 0.0)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(Float64(im * im) / Float64(t_0 - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(t_0 + re))));
	end
	return tmp
end

function tmp_2 = code(re, im)
	t_0 = sqrt(((re * re) + (im * im)));
	tmp = 0.0;
	if (re < 0.0)
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	end
	tmp_2 = tmp;
end

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[re, 0.0], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(im * im), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(t$95$0 + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sqrt{re \cdot re + im \cdot im}\\
\mathbf{if}\;re < 0:\\
\;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t_0 - re}}\right)\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t_0 + re\right)}\\


\end{array}
\end{array}

Reproduce

?
herbie shell --seed 2023350 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))